Modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules,and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated,and several results are generalized from the affine Lie algebras. In particular,imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.     

Generalized Verma modules over some Block algebras

In this paper, a class of generalized Verma modules M(V) over some Block type Lie algebra ℬ(G) are constructed, which are induced from nontrivial simple modules V over a subalgebra of ℬ(G). The irreducibility of M(V) is determined.      

某些$W$-型李代数的Verma模

本文我们研究了两类$W$-型李代数$g(\lambda)$的Verma模的结构. 在一定条件下,我们决定了这些Verma模的可约性及相应的奇异向量.     

On multiplicities of simple subquotients in generalized Verma modules

We reduce the problem on multiplicities of simple subquotients in an -stratified generalized Verma module to the analogous problem for classical Verma modules.     

Gelfand-Kirillov Dimension and Reducibility of Scalar Generalized Verma Modules

The Gelfand-Kirillov dimension is an invariant which can measure the size of infinite-dimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case.     

Verma Modules Over Generalized Block Algebras B(b (1), b (2))

ABSTRACT

Given a field F with characteristic zero, a free Abelian group G with rank two, and a total order ? on G which is compatible with the addition, we define Verma modules M([ddot], ?) over the generalized Block algebra B(b ⁽¹⁾, b ⁽²⁾) with b ⁽¹⁾, b ⁽²⁾ ∈ F. The irreducibility of the module M([ddot], ?) is completely determined in this article.     

Generalized imaginary Verma and Wakimoto modules

We develop a general technique of constructing new irreducible weight modules for any affine Kac-Moody algebra using the parabolic induction, in the case when the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. Our approach unifies and generalizes all previously known results with imposed restrictions on inducing modules. We also define generalized imaginary Wakimoto modules which provide an explicit realization for generic generalized imaginary Verma modules.     

Submodule Structure of Generalized Verma Modules Induced from Generic Gelfand-Zetlin Modules

For complex Lie algebra sl(n, C) we study the submodule structure of generalized Verma modules induced from generic Gelfand-Zetlin modules over some subalgebra of type sl(k, C). We obtain necessary and sufficient conditions for the existence of a submodule generalizing the Bernstein-Gelfand-Gelfand theorem for Verma modules.     

Reducibility of generalized Verma modules for Hermitian symmetric pairs

In this article, we give some practical criteria to determine the reducibility of generalized Verma modules (induced from finite-dimensional modules) in the Hermitian symmetric case. Our criteria are given by the information of the corresponding highest weights of the finite-dimensional modules and relatively easy to verify. This article is inspired by earlier work of Kubo about Jantzen's criterion.     

Verma modules for twisted Yangians

Abstract

A sufficient and necessary condition for the reducibility of the Verma modules over the twisted Yangians of rank one is given.     

Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

     

Conformally invariant systems of differential operators

Kostant's theory of conformally invariant differential operators on certain homogeneous spaces is generalized to cover conformally invariant systems of endomorphism-valued differential operators. In particular, the connection discovered by Kostant between conformally invariant operators and highest weight vectors in generalized Verma modules is extended.     

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.     

Structure of T Modules and Restricted Duals: The Classical and the Quantum Case

Abstract

A concrete realization of Enright's T modules is obtained. This is used to show their self-duality. As a consequence, the restricted duals of Verma modules are also identified.     

Representation theory of N=2 super Virasoro algebra: twisted sector

In this article, we study the structure of Verma modules and Fock modules over the twisted sector of the N=2 super Virasoro algebras.     

Quantum Deformations of α-Stratified Modules

We construct quantum analogues of a class of generalized Verma modulesinduced from nonsolvable parabolic subalgebras of simple Lie algebras. Weshow that these quantum modules are true deformations of the underlyingclassical modules in the sense that the weight-space decomposition ispreserved.     

带三角分解李代数的赋值模

设F是特征零的域，L是F上的带三角分解的李代数，L^-是相应的Loop代数．本文将定义L^-上赋值模的概念，并给出其不可约模的张量积是不可约模的等价条件．     

一般线性Lie超代数的微分算子表示

对一般线性Lie超代数的Verma模及向量相干态(vector-coherent-states)表示给出了以向量为系数的微分算子实现.对一般线性Lie超代数的单一非典型(singlyatypical)Kac模,我们具体写出了本原权向量,并对这种情形确切写出了用以刻画向量相干态表示的单子表示的微分方程.